Giga-Gauss Scale Quasistatic Magnetic Field Generation with Laser

Giga-Gauss scale quasistatic magnetic ﬁeld generation with laser

Ph.Korneev∗ NRNU MEPhI, Moscow 115409, Russian Federation and University of Bordeaux, CNRS, CEA, CELIA, 33405 Talence, France

E. D’Humi`eres, V. Tikhonchuk University of Bordeaux, CNRS, CEA, CELIA, 33405 Talence, France

A simple setup for the generation of ultra-intense quasistatic magnetic fields is proposed and analysed. Estimations and numerical Particle-In-Cell calculations show that magnetic fields of gigagauss scale may be generated with conventional powerful relativistic lasers interacting with the appropriate targets of a special geometry. The setup may be useful for a wide range of applications, from laboratory astrophysics to magnetized ICF schemes. Keywords: Magnetic field, laboratory astrophysics, θ−pinch, laser-plasma interaction.

INTRODUCTION simulations, then present a brief analysis of the magnetic field structure and estimate physical parameters for the Magnetic fields of different scales have been the sub- possible experimental studies. Finally, we conclude by ject of many studies since their discovery hundreds of briefly discussing possible applications. years ago. It was then found, that astrophysical phenomena possess magnetic fields with amplitudes within a AN EXAMPLE FOR A MAGNETIC FIELD huge range from microgausses up to teragausses and even GENERATION IN AN ’ESCARGOT’ TARGET greater, deep in the relativistic region. The ultrahigh magnetic fields generation in the laboratory is a modern Electrons, accelerated forward by a relativistic laser, trend which includes the astrophysical modelling, rela- may generate magnetic fields in Z-pinch geometry, which tivistic studies with atoms and particles, Inertial Con- lives as long as the current of the accelerated particles finement Fusion (ICF), etc. Along with the utilization lives. The magnetic field strength may be high in a case of of the pulsing and exploding magnetic field generators, high currents, but the spacial and temporary properties laser-assisted generation of magnetic fields attracts great of these fields narrow their subsequent utilization. In interest. Modern laser facilities are powerful instruments the opposite, pulsed currents in a solenoid geometry [2] for the generation of intense magnetic fields (see, i.e. [1– produce relatively long living localized magnetic fields in 3]), as they may concentrate a lot of electromagnetic en- θ-pinch geometry. Our idea is to produce the relativistic ergy in small time and space regions. solenoid-like current of ionized electrons, for this we make In the present letter we propose a novel mechanism use of surface electron guiding effect [4], and the plasma for the generation of the intense magnetic field, based on mirror effect [8], with a mirror of a prescribed geometry. the direct ponderomotive electrons acceleration by a laser For the current study we choosed a helix, or ’escargot’, pulse applied to a target of a special geometry. Electron target, which is closed to a cylindrical geometry, but has acceleration mechanism is close to that described in [4], a hole, for the entrance of a laser pulse, see Fig.1(A1). and is accompanied with the electron guiding along the surface, irradiated by an oblique incident intense laser δr θ r(θ) = r0 1 + , θ ∈ (0, 2π) , (1) pulses. The effect was experimentally confirmed [5], with r0 2π the laser intensity 1..2×1018W/cm2, and studied numer- ically in different geometries [6,7]. It was noticed, that where θ = 0 corresponds to the upper direction of the accelerated electrons produce time-dependent currents, vertical axis in Fig.1. We examine laser-target interac- which may form self-consistent structures with the corre- tion with our specific target with 2D3V Particle-In-Cell sponding magnetic field. As we show below, the surface code PICLS [9]. To prove the robustness of the mecha- arXiv:1409.5246v1 [physics.plasm-ph] 18 Sep 2014 electron guiding effect may be used in quasistatic high nism, we presented several runs with different target and magnetic field generation, with the field amplitudes of laser parameters. the order of giga-gauss level, and characteristic times of To the sake of brevity, we detaily describe only one the order of at least several picoseconds. This mechanism of the runs (run (a)). In it the laser intensity was 5 × 19 2 demands, that laser intensity posesses relativistic values, 10 W/cm , the laser pulse with a wavelength of 0.93 and a sufficient energy deposition for achieving high am- microns had a duration of 500fs. The target itself was plitudes of the produced magnetic fields. Declaring the defined by (1), with r0 = 43 microns, δr = 28 microns, principal possibility of the setup for a laser-assisted in- and it was composed with two layers by the material (1) tense magnetic field production, with the numerical mod- with ion charge Z = 79, ion masses mi = 36169me (2) elling we show examples of the generation and the scale of the inner 1 micron width layer, and mi = 72338me of the generated fields. We organize the letter as follow- of the outer 2 microns width layer (sublayer). The ion 21 −3 ing: first, we present the results of the Particle-In-Cell density was ni = 10 cm . Electrons with masses me 2

Energy balance 1.6 total energy 1.4 e-m energy energy of heavy ions 1.2 energy of Au ions electron energy 1

0.8

0.6 Energy, rel. units 0.4

0.2

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Time, picoseconds

FIG. 2: Energy balance during the interaction, shown in Fig. 1 (run (a)). After 2 picoseconds, when the laser pulse is gone, electromagnetic energy is composed only by the magnetic ﬁeld energy, which has the order of 5 − 7% of the maximum (total laser pulse) energy.

sufficient part of it is guided by the walls of the ’escargot’ chamber, which serve as plasma mirrors of the specified form. Electrons, ablated during the laser pulse reflection from the target inner surface, are accelerated by a laser ponderomotive force along the surface, due to the surface guided mechanism, and produce a strong current, which generates a magnetic field in z−direction (normal to the figures plane). The generated Bz field do not allow electrons to escape from the inner region of the target, forming an initial θ−pinch-like structure. As time goes, TNSA mechanism [10] of ion acceleration comes into play, pulling ions from the target inner surface into the inner region. This secondary effect leads to stabilization of the θ−pinch-like structure, and at the same time compresses the initial magnetic field. To estimate a conversion efficiency from laser to the FIG. 1: Electron density (panels A1-D1), and Bz field (panels θ−pinch magnetic structure energy, we consider energy A2-D2) at different time moments: 0.62, 1.9, 3.1, 4.3 ps cor- balance during the interaction in Fig.2. The total en- respondingly for A1,B1,C1,D1 and A2,B2,C2,D2, for the run ergy is increasing almost linearly with time, when laser (a). Electron density is shown in the units of 1.3 × 1021cm−3, and is cut on the value of 6.5 × 1021cm−3 magnetic field is is on, and decreases after the laser turns off. This energy shown in the units of 1.16×108Gauss, so that maximum value loss comes mainly from the energetic electrons, which of 2.6 in the colorbar corresponds to 3 × 108Gauss. leave the simulation box. Electromagnetic energy ini- tially also grows linearly, but approximately at 0.4 picoseconds, when the laser starts to interact with the inner had the density ne = Zni. The reduced ion density in surface of the target, it is absorbed by electrons. Later this run corresponds to a possible preplasma layer on the on, during the laser propagation, electron energy grows inner surface of the target, and the more massive sublayer with time, reaching the value of the order of ≈ 60% of was introduced to save the calculation time but at the the total energy at the end of the laser pulse (about 1.5 same time decrease the expansion time. The matter was picoseconds). In contrary, electromagnetic energy stops presented as fully ionized cold ions, 1 particle per a cell, to grow substantially, but after the laser is switched off and 100keV hot electrons, 79 particles per a cell. The it does not go to the zero value. This small part, left simulation box was 2160 × 2304 cells, or approximately after 2 picoseconds, corresponds to the energy of the de- 100 × 110 microns. The resolution was 20 steps in a time veloped magnetic field structure. From Fig.2 it may be unit and 20 in a space unit. The process of the interaction concluded, that after the interaction, the magnetic en- is shown in Fig.1, by electron density and magnetic field ergy contains about 5−7% of the total laser pulse energy. Bz at several subsequent time moments. We examined the robustness of the proposed mecha- As it is seen from the results of the PIC simulations in nism by several runs with different laser and target pa- Fig.1, even if there is a great laser energy absorption, a rameters, and found the same effect. As an example we 3

structures may be formed inside the cavity. Topologically in our case they may be similar to the θ−pinch, which in a stationary situation can be described by the equation of the pressure balance

B2 z + P ≈ 0, (2) 8π where P is the pressure of the plasma. When the inner walls of the target are substantially heated by a laser pulse, the material of the walls ablates and an interest- FIG. 3: Electron density (left panel), and Bz field (right ing effect takes place: the ablation pressure starts to panel) at the time moment 6.36 ps, for the run (b). Elec- compress the inner magnetic structure with hot dense tron density is cut on the level of 8 × 1021cm−3, magnetic plasma. This hot plasma can only slowly be mixing with field is shown in the units of 1.16 × 108Gauss. the magnetic field because it rapidly becomes colisionless. However, if the absorption of the laser pulse occurs before show the electron density and the magnetic field Bz at the whole inner target surface is heated, a cold part of the a late time 6.36 picoseconds, in Fig.3. For this run surface may be magnetized. In Fig.1(C1,C2) it is seen, (b), the parameters we used are: the laser intensity was that to the right of the center of the target, where laser 1020W/cm2, the laser wavelength was 0.93 microns, the energy is high enough to produce hot plasma, magnetic laser pulse duration was 500fs. The target with r0 = 100 field is separated from the surface plasma. In contrary, microns and δr = 50 microns was composed by the mate- to the left of the center of the target, the surface plasma (1) is being more magnetized during its cold stage because rial with ion charge Z = 79, ion masses mi = 36169me, 22 −3 it can absorb enough laser energy to become collisionless the ion density was ni = 2 × 10 cm . The size of the target was approximately 2 times larger, than in the run later in time. So, on the second stage, the hot ablated (a). As it is seen in Fig.3, the scale of the produced mag- plasma comes to the equilibrium (2) with the magnetic netic field and the magnetic structure are quite similar field inside the target hollow. During this stage, one may to that for run (a). estimate the magnetic field from the condition, that pres- 2 sure Bz /8π is the same as the pressure P = neTe. For 21 −3 the parameters from Fig.1 for run (a), ne ∼ 10 cm DISCUSSION and Te ∼ 1MeV – the ponderomotive energy for the con- 9 sidered laser pulse, one can get Bz ∼ 0.2 × 10 Gauss, in The laser-target interaction and magnetic field gener- accordance with the value in Fig.1(D2). ation may be divided in several important steps, such as The correct scaling of the presented estimations for electron currents generation, field formation, and mag- the both stages, shows that the magnetic field amplitude netic structure development inside a hot cavity. The is defined mainly by the laser pulse parameters. This acceleration of surface electrons, lasts in run (a) from also comes out from the comparison of different runs, the beginning of the laser-surface interaction up to the as run (a) and run (b). The reason for this lies in the time, when the laser pulse energy is almost totally ab- fact, that the ruling parameter of the irradiated solid is sorbed in electron surface plasma, around 500 − 600 fs, its electron density, which appears to be overcritical for Fig.1(B1,B2). During this initial stage, the magnetic different targets materials. The last may define, however, field strength may be roughly estimated by surface elec- the life time of the generated magnetic field structure and tron currents, on the base of Ampère’slaw. Let κ be its geometrical properties. the conversion ratio from laser energy to energy in hot We considered topologically unclosed target (1), which electrons, moving along the inner surface of the target, does not allow real solenoidal currents. As we see in Fig.1 which normally is of the order of 0.1 [4]. For relativistic and Fig.3, in this case the generated magnetic structure intensities electron velocity is close to the light velocity, has a dipole-like geometry, as it is defined by the surface then electron currents. When the laser pulse inside the hollow is intense enough, though it loses a sufficient part of ∗ Bz ∼ 4πκencr , its energy during the reflection, it produces electron currents, which increase negative charge of the rest of the where e is the electron charge, nc is the critical electron target, the left down part in our case. Return currents density, r∗ is the characteristic length of the surface cur- then appears along the target surface, on the time scale 21 −3 ∗ rent. Considering nc ∼ 10 cm , r ∼ 10 microns, we of r0/c, appear where c is the light velocity, and they be- 9 can estimate the generated field as Bz ∼ 0.6×10 Gauss, come responsible for the magnetic field generation with the order corresponds to Fig.1(B2). the opposite direction. For the target sizes considered For the later stage, we mention, that depending on here, the time scale of the returning currents are of the initial conditions, namely details of a target geometry order of 1 ps, which is also seen in Fig.1. For the gen- and material, laser parameters, etc., different magnetized eration of the monopole-type magnetic field, connected 4

in a variety of applications, such as laboratory astrophysics experiments, neutron production, different aspects of ICF, such as, i.e. electron magnetic collimation [12], etc. We discuss here several possibilities. In laboratory astrophysics applications, an intense quasistationary magnetic field generation along with the production of a highly magnetized plasma may be used, for example, for a b c the studies of a magnetic reconnection phenomena, see, i.e. a recent review [13]. Depending on laser and target parameters it may be possible to achieve magnetized FIG. 4: Examples of target geometries for the experimen- plasmas, propagating in the opposite directions, with dif- tal applications of the considered effect: a – two cone-like ’escargot’ targets for collisions of magnetized plasmas; b – ferent orientations of the magnetic field. For the stan- magnetic trap geometry; c – microtocamak geometry. Black dard reconnection geometry with magnetic fields in two arrows show laser pulses directions. colliding plasmas of the opposite directions, a possible example of a target is shown in Fig.4a. However, already inside a single ’escargot’ target (1), with adjusted laser currents must be generated. This may become possible and target paramemeters, reconnection phenomena may in 3D geometry, see, i.e. Fig.4 b and c. be looked for, as it follows from the structure of the gen- Considering different parameters of the interaction erated magnetic field. Another interesting application with the ’escargot’ target, we found the proposed scheme may be micrometer-scale magnetic traps, see Fig.4b and is in general robust. However, at least two competing c. Target sizes and magnetic field values may be of the optimizations are possible. The one is an increase of the order of magnitude of interest for neutron production or inner target radii to increase the entrance aperture, the even magnetized fusion schemes [14]. For instance, if the laser pulse inner reflection, the symmetry of the produced field values are of the order of 100MGauss, and the trap surface currents, and the energy conversion from the laser radius is about 50 microns, it can contain protons with to magnetic field. It was shown earlier [6,7], that surface energies ∼ 30 MeV, and α−particles with energies of the currents may be effectively generated when laser incident order of ∼ 10 MeV. With the advanced target production angle is about 50-70 degrees. Too small target size may technologies, it may become possible to generate toroidal result in an increased laser energy absorption, without magnetic structures, i.e. in target, shown in Fig.4c. efficient directed electron currents generation. The other In conclusion, the idea described in the present let- optimization is the decreasing of the inner target radii ter is based on the ponderomotive acceleration of elec- to increase the concentration of the magnetic field en- trons along a target- predefined trajectory. It may work ergy and to perform a stage of additional compression of if laser intensity exceeds relativistic values. Accelerated the generated magnetic field by the ablated plasmas from electrons produce currents, which form a self-consistent the inner target surface. The defining parameter for the time-dependent structure with the corresponding mag- target size may thus become an entrance aperture, since netic field. As a first ”proof-of-principle” example, we it is defined by an experimental technique. In the two show that to form a simple long-living θ−pinch type elec- presented runs (a) and (b) the target size was chosen tromagnetic structure, an ’escargot’-like target may be to be close to the optimum between the two considered used. More complex magnetic field microstructures may compeeting tendencies. be generated with certain target geometries.

CONCLUSIONS AND PERSPECTIVES ACKNOWLEDGMENTS

We presented two-dimentional calculations for the Authors greatly appreciate usefull discussions with high quasistationary magnetic field generation mecha- S.Fujioka and J.Santos. The work is in part supported nism with intense laser pulses. For the possible exper- by the French National funding agency ANR within the imental realization, it is nesessary to understand the role project SILAMPA. of 3D effects. In our simulations the intensity was of the order of 1019..1020 W/cm2, and the pulse duration was 0.5 ps. In terms of the laser energies, it corresponds to the order of 100 J. This is the moderate modern level ∗ scale, which makes an experimental realization feasible. Electronic address: [email protected] However, more powerfull facilities, such as, i.e. upcoming [1] D.D. Ryutov. Using intense lasers to simulate aspects of accretion discs and outflows in astrophysics. Astrophysics PETAL [11], may allow the production of more intense and Space Science, 336(1):21–26, 2011. magnetic fields. [2] Shinsuke Fujioka, Zhe Zhang, Kazuhiro Ishihara, The proposed scheme for the magnetic field genera- Keisuke Shigemori, Youichiro Hironaka, Tomoyuki Jo- tion, with the correspondent modifications, may be used hzaki, Atsushi Sunahara, Naoji Yamamoto, Hideki 5

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